:: Algebraic Numbers
:: by Yasushige Watase
::
:: Received December 15, 2016
:: Copyright (c) 2016-2018 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, FINSEQ_1, SUBSET_1, FUNCT_1, ARYTM_3, TARSKI, NAT_1,
XBOOLE_0, SUPINF_2, ZFMISC_1, GROUP_1, STRUCT_0, POLYNOM1, POLYNOM2,
C0SP1, REALSET1, ARYTM_1, RELAT_1, CARD_1, XXREAL_0, VECTSP_1, ALGSTR_0,
FUNCT_7, AFINSQ_1, CARD_3, MESFUNC1, POLYNOM3, FUNCSDOM, ORDINAL4, INT_2,
GAUSSINT, BINOP_2, COMPLFLD, EC_PF_1, XCMPLX_0, FINSEQ_2, INT_3,
ALGSEQ_1, POLYNOM4, PARTFUN1, IDEAL_1, CARD_FIL, VECTSP_2, POLYNOM5,
HURWITZ, RATFUNC1, MSSUBFAM, QUOFIELD, BINOP_1, RING_1, RING_2, LATTICES,
GCD_1, RLVECT_1, ALGNUM_1, RAT_1, INT_1, WAYBEL_8;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, XTUPLE_0, MCART_1, DOMAIN_1,
ORDINAL1, RELAT_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, NUMBERS, INT_1,
FUNCOP_1, BINOP_1, FUNCT_4, FUNCT_7, SETWISEO, FINSEQ_1, FINSEQ_2,
XCMPLX_0, XXREAL_0, CARD_1, XREAL_0, PBOOLE, VALUED_0, NAT_1, NAT_D,
RAT_1, NEWTON, BINOP_2, MEMBERED, STRUCT_0, ALGSTR_0, C0SP1, NORMSP_1,
VFUNCT_1, VECTSP_2, ALGSEQ_1, ALGSTR_1, RLVECT_1, GROUP_1, VECTSP_1,
RINGCAT1, RING_3, GROUP_6, RING_2, POLYNOM1, UPROOTS, HURWITZ, RATFUNC1,
BINOM, INT_3, GCD_1, RVSUM_1, COMPLFLD, POLYNOM3, POLYNOM4, POLYNOM5,
FVSUM_1, PRE_POLY, REALSET1, COMPLEX1, EC_PF_1, GAUSSINT, IDEAL_1,
RING_1, MSSUBFAM, QUOFIELD, RING_4;
constructors TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, FUNCT_1, RELSET_1, NUMBERS,
ORDINAL1, FUNCT_2, FINSEQ_1, FINSEQOP, STRUCT_0, ALGSTR_0, C0SP1,
VECTSP_1, NORMSP_1, VFUNCT_1, POLYNOM3, SETWISEO, BINOP_1, REAL_1,
RFINSEQ, FINSOP_1, BINARITH, VECTSP_2, GRCAT_1, REALSET2, QUOFIELD,
ALGSTR_1, POLYNOM4, POLYNOM5, NAT_D, FVSUM_1, ALGSEQ_1, FUNCT_7, BINOP_2,
INT_1, RLVECT_1, GROUP_1, RINGCAT1, MOD_4, GROUP_6, RING_3, RING_2,
GROUP_4, FINSEQ_4, MATRIX_1, POLYNOM1, UPROOTS, HURWITZ, POLYNOM2,
RING_4, XXREAL_2, XTUPLE_0, PARTFUN1, RVSUM_1, XCMPLX_0, RAT_1, NEWTON,
VALUED_0, MEMBERED, NAT_1, FINSEQ_2, RATFUNC1, BINOM, GCD_1, IDEAL_1,
EC_PF_1, GAUSSINT, AFINSQ_2, FUNCT_4, REALSET1, MSSUBFAM;
registrations ALGSTR_0, GAUSSINT, CARD_1, FINSEQ_1, INT_3, FINSEQ_2, FUNCT_1,
FUNCT_2, INT_1, MEMBERED, NAT_1, NUMBERS, ORDINAL1, POLYNOM3, POLYNOM5,
REALSET1, RELAT_1, RING_2, STRUCT_0, VECTSP_1, XREAL_0, RATFUNC1, RAT_1,
XXREAL_0, COMPLFLD, XCMPLX_0, VALUED_0, POLYNOM4, ALGSTR_1, RLVECT_1,
RING_1, RING_4, IDEAL_1, XBOOLE_0, RINGCAT1, GROUP_6, GROUP_1, GROUP_4,
POLYNOM2, QUANTAL1, FUNCSDOM, GCD_1, RELSET_1;
requirements NUMERALS, BOOLE, SUBSET, ARITHM, REAL;
begin :: Preliminaries
reserve i,j for Nat;
reserve A,B for Ring;
theorem :: ALGNUM_1:1
for L1,L2,L3 be Ring st L1 is Subring of L2 & L2 is Subring of L3 holds
L1 is Subring of L3;
theorem :: ALGNUM_1:2
F_Rat is Subfield of F_Complex;
theorem :: ALGNUM_1:3
F_Rat is Subring of F_Complex;
theorem :: ALGNUM_1:4
INT.Ring is Subring of F_Complex;
theorem :: ALGNUM_1:5
for x, y be Element of B, x1, y1 be Element of A st A is Subring of B &
x = x1 & y = y1 holds x + y = x1 + y1;
theorem :: ALGNUM_1:6
for x, y be Element of B, x1, y1 be Element of A st
A is Subring of B & x = x1 & y = y1 holds x * y = x1 * y1;
registration
let c be Complex;
reduce In(c,F_Complex) to c;
end;
begin
:: Define Extended eval Function for commutative rings A c= B
:: based upon POLYNOM4
definition
let A,B be Ring;
let p be Polynomial of A;
let x be Element of B;
func Ext_eval(p,x) -> Element of B means
:: ALGNUM_1:def 1
ex F be FinSequence of B st it = Sum F & len F = len p
& for n be Element of NAT st n in dom F holds
F.n = In(p.(n-'1),B) * (power B).(x,n-'1);
end;
theorem :: ALGNUM_1:7
for n being Element of NAT, A,B be Ring, z be Element of A st
A is Subring of B holds
(power B).(In(z,B),n) = In((power A).(z,n),B);
theorem :: ALGNUM_1:8
for x1,x2 be Element of A st A is Subring of B holds
In(x1,B) + In(x2,B) = In(x1+x2,B);
theorem :: ALGNUM_1:9
for x1,x2 be Element of A st A is Subring of B holds
In(x1,B) * In(x2,B) = In(x1*x2,B);
theorem :: ALGNUM_1:10
for F be FinSequence of A,
G be FinSequence of B
st A is Subring of B & F = G holds In(Sum F,B) = Sum G;
theorem :: ALGNUM_1:11
for n be Nat, x be Element of A, p be Polynomial of A st A is Subring of B
holds In(p.(n-'1),B)*(power B).(In(x,B),n-'1)
= In(p.(n-'1) * (power A).(x,n-'1),B);
theorem :: ALGNUM_1:12
for x be Element of A, p be Polynomial of A st A is Subring of B holds
Ext_eval(p,In(x,B)) = In(eval(p,x),B);
:: Modify POLYNOM4:17
theorem :: ALGNUM_1:13
for x be Element of B holds Ext_eval(0_.A,x) = 0.B;
:: Modify POLYNOM4:18
theorem :: ALGNUM_1:14
for A,B being non degenerated Ring
for x be Element of B st A is Subring of B holds
Ext_eval(1_.A,x) = 1.B;
:: Modify POLYNOM4:19
theorem :: ALGNUM_1:15
for x be Element of B, p,q be Polynomial of A st A is Subring of B holds
Ext_eval(p+q,x) = Ext_eval(p,x) + Ext_eval(q,x);
theorem :: ALGNUM_1:16
for p,q be Polynomial of A st A is Subring of B & len p > 0 & len q > 0
for x be Element of B holds
Ext_eval((Leading-Monomial(p))*'(Leading-Monomial(q)),x)
= In(p.(len p-'1)*q.(len q-'1),B)*(power B).(x,len p+len q-'2);
theorem :: ALGNUM_1:17
for p be Polynomial of A for x be Element of B st A is Subring of B holds
Ext_eval(Leading-Monomial(p),x)
= In(p.(len p-'1),B) * (power B).(x,len p-'1);
::Modify POLYNOM_4:Lm3:
theorem :: ALGNUM_1:18
for B be comRing
for p,q be Polynomial of A for x be Element of B st A is Subring of B holds
Ext_eval( (Leading-Monomial p)*'(Leading-Monomial q),x) =
Ext_eval(Leading-Monomial(p),x)*Ext_eval(Leading-Monomial(q),x);
:: Modify POLYNOM4:23
theorem :: ALGNUM_1:19
for B be comRing
for p,q be Polynomial of A for x be Element of B st A is Subring of B holds
Ext_eval((Leading-Monomial p)*'q,x)
= Ext_eval(Leading-Monomial(p),x) * Ext_eval(q,x);
:: Modify POLYNOM4:24
theorem :: ALGNUM_1:20
for B be comRing
for p,q be Polynomial of A for x be Element of B st A is Subring of B
holds Ext_eval(p*'q,x) = Ext_eval(p,x) * Ext_eval(q,x);
:: modified POLYNOM5:37
theorem :: ALGNUM_1:21
for x be Element of B, z0 be Element of A
st A is Subring of B holds Ext_eval(<%z0%>,x) = In(z0,B);
:: modified POLYNOM5:44
theorem :: ALGNUM_1:22
for x be Element of B, z0,z1 be Element of A st A is Subring of B
holds Ext_eval(<%z0,z1%>,x) = In(z0,B)+In(z1,B)*x;
begin
:: Definition of Integral Element over A in B
definition
let A,B be Ring;
let x be Element of B;
pred x is_integral_over A means
:: ALGNUM_1:def 2
ex f be Polynomial of A st LC f = 1.A & Ext_eval(f,x) = 0.B;
end;
theorem :: ALGNUM_1:23
for A being non degenerated Ring
for a be Element of A st A is Subring of B holds
In(a,B) is_integral_over A;
definition
let A be non degenerated Ring, B be Ring;
assume
A is Subring of B;
func integral_closure(A,B) -> non empty Subset of B equals
:: ALGNUM_1:def 3
{z where z is Element of B: z is_integral_over A};
end;
definition
let c be Complex;
attr c is algebraic means
:: ALGNUM_1:def 4
ex x being Element of F_Complex st x = c & x is_integral_over F_Rat;
end;
definition
let x be Element of F_Complex;
redefine attr x is algebraic means
:: ALGNUM_1:def 5
x is_integral_over F_Rat;
end;
definition
let c be Complex;
attr c is algebraic_integer means
:: ALGNUM_1:def 6
ex x being Element of F_Complex st x = c & x is_integral_over INT.Ring;
end;
definition
let x be Element of F_Complex;
redefine attr x is algebraic_integer means
:: ALGNUM_1:def 7
x is_integral_over INT.Ring;
end;
notation
let x be Complex;
antonym x is transcendental for x is algebraic;
end;
registration
cluster rational -> algebraic for Complex;
end;
registration
cluster algebraic for Complex;
cluster algebraic for Element of F_Complex;
end;
registration
cluster integer -> algebraic_integer for Complex;
end;
registration
cluster algebraic_integer for Complex;
cluster algebraic_integer for Element of F_Complex;
end;
definition
let A,B be Ring;
let x be Element of B;
func Ann_Poly(x,A) -> non empty Subset of Polynom-Ring A equals
:: ALGNUM_1:def 8
{p where p is Polynomial of A: Ext_eval(p,x) = 0.B};
end;
theorem :: ALGNUM_1:24
for A,B be Ring, w be Element of B, x, y being Element of Polynom-Ring A
st A is Subring of B & x in Ann_Poly(w,A) & y in Ann_Poly(w,A)
holds x + y in Ann_Poly(w,A);
theorem :: ALGNUM_1:25
for B be comRing, z be Element of B, p, x being Element of Polynom-Ring A
st A is Subring of B & x in Ann_Poly(z,A) holds p * x in Ann_Poly(z,A);
theorem :: ALGNUM_1:26
for B be comRing
for w be Element of B, p, x being Element of Polynom-Ring A
st A is Subring of B & x in Ann_Poly(w,A) holds x * p in Ann_Poly(w,A);
theorem :: ALGNUM_1:27
for A be non degenerated Ring
for B be non degenerated comRing
for w be Element of B st A is Subring of B
holds Ann_Poly(w,A) is proper Ideal of Polynom-Ring A;
begin :: Properties of Polynomial Ring over PID.
reserve K, L for Field;
theorem :: ALGNUM_1:28
for K,L be Field, w be Element of L st K is Subring of L holds
ex g be Element of Polynom-Ring K st {g}-Ideal = Ann_Poly(w,K);
theorem :: ALGNUM_1:29
for K,L be Field, z be Element of L st z is_integral_over K
holds Ann_Poly(z,K) <> {0.Polynom-Ring K};
theorem :: ALGNUM_1:30
for K be Field, p be Element of Polynom-Ring K st p <> 0_.K holds
p is non zero Element of the carrier of Polynom-Ring K;
theorem :: ALGNUM_1:31
for K,L be Field, w be Element of L st K is Subring of L
holds Ann_Poly(w,K) is quasi-prime;
theorem :: ALGNUM_1:32
for K,L be Field, w be Element of L st K is Subring of L &
w is_integral_over K holds Ann_Poly(w,K) is prime;
theorem :: ALGNUM_1:33
for K,L be Field, z be Element of L st K is Subring of L &
z is_integral_over K
ex f be Element of Polynom-Ring K
st f <> 0_.K & {f}-Ideal = Ann_Poly(z,K) & f = NormPolynomial(f);
theorem :: ALGNUM_1:34
for K,L be Field, z be Element of L,f,g be Element of Polynom-Ring K st
z is_integral_over K &
{f}-Ideal = Ann_Poly(z,K) & f = NormPolynomial(f) &
{g}-Ideal = Ann_Poly(z,K) & g = NormPolynomial(g)
holds f = g;
definition
let K,L be Field;
let z be Element of L;
assume that
K is Subring of L and
z is_integral_over K;
func minimal_polynom(z,K) -> Element of the carrier of Polynom-Ring K
means
:: ALGNUM_1:def 9
it <> 0_.K & {it}-Ideal = Ann_Poly(z,K) & it = NormPolynomial(it);
end;
definition
let K,L be Field;
let z be Element of L;
assume that
K is Subring of L and
z is_integral_over K;
func deg_of_integral_element(z,K) -> Element of NAT equals
:: ALGNUM_1:def 10
deg (minimal_polynom(z,K));
end;
definition
let A,B be Ring;
let x be Element of B;
func hom_Ext_eval(x,A) -> Function of Polynom-Ring A,B means
:: ALGNUM_1:def 11
for p be Polynomial of A holds it.p = Ext_eval(p,x);
end;
registration
let x be Element of F_Complex;
cluster hom_Ext_eval(x,F_Rat) -> unity-preserving additive multiplicative;
end;
theorem :: ALGNUM_1:35
for x be Element of F_Complex holds
F_Complex is (Polynom-Ring F_Rat)-homomorphic;
theorem :: ALGNUM_1:36
for x be Element of B, z be object st z in rng hom_Ext_eval(x,A)
holds z in B;
definition
let x be Element of F_Complex;
func FQ(x) -> Subset of F_Complex equals
:: ALGNUM_1:def 12
rng hom_Ext_eval(x,F_Rat);
end;
registration
let x be Element of F_Complex;
cluster FQ(x) -> non empty;
end;
theorem :: ALGNUM_1:37
for x,z1,z2 be Element of F_Complex st
z1 in FQ(x) & z2 in FQ(x) holds z1 + z2 in FQ(x);
theorem :: ALGNUM_1:38
for x,z1,z2 be Element of F_Complex st
z1 in FQ(x) & z2 in FQ(x) holds z1 * z2 in FQ(x);
theorem :: ALGNUM_1:39
for x be Element of F_Complex, a be Element of F_Rat holds
a in FQ(x);
definition
let x be Element of F_Complex;
func FQ_add(x) -> BinOp of FQ(x) equals
:: ALGNUM_1:def 13
addcomplex || FQ(x);
end;
definition
let x be Element of F_Complex;
func FQ_mult(x) -> BinOp of FQ(x) equals
:: ALGNUM_1:def 14
multcomplex || FQ(x);
end;
theorem :: ALGNUM_1:40
for x be Element of F_Complex, z, w be Element of FQ(x) holds
(FQ_add(x)).(z,w) = z+w;
theorem :: ALGNUM_1:41
for x be Element of F_Complex, z, w be Element of FQ(x) holds
(FQ_mult(x)).(z,w) = z*w;
theorem :: ALGNUM_1:42 :::?????
for x be Element of F_Complex holds
In(1.F_Complex, FQ(x)) = 1.F_Complex;
theorem :: ALGNUM_1:43
In(-1.F_Rat,F_Complex) = -1.F_Complex;
definition
let x be Element of F_Complex;
func FQ_Ring(x) -> strict non empty doubleLoopStr equals
:: ALGNUM_1:def 15
doubleLoopStr(# FQ(x), FQ_add(x), FQ_mult(x),In(1.F_Complex,FQ(x)),
In(0.F_Complex,FQ(x)) #);
end;
theorem :: ALGNUM_1:44
for x be Element of F_Complex holds FQ_Ring(x) is Ring;
registration
let x be Element of F_Complex;
cluster FQ_Ring(x) -> Abelian add-associative right_zeroed
right_complementable associative well-unital distributive;
end;
registration
let z be Element of F_Complex;
cluster FQ_Ring(z) -> domRing-like commutative non degenerated;
end;
theorem :: ALGNUM_1:45
for x be Element of F_Complex holds
[:RAT,RAT:] c= [:FQ(x),FQ(x):] & [:FQ(x),FQ(x):] c= [:COMPLEX,COMPLEX:];
theorem :: ALGNUM_1:46
for x be Element of F_Complex holds
the addF of F_Rat = (the addF of FQ_Ring(x))||RAT;
theorem :: ALGNUM_1:47
for x be Element of F_Complex holds
the multF of F_Rat = (the multF of FQ_Ring(x))||RAT;
theorem :: ALGNUM_1:48
for x be Element of F_Complex holds F_Rat is Subring of FQ_Ring(x);
theorem :: ALGNUM_1:49
for f,g be Element of Polynom-Ring K st
f <> 0.Polynom-Ring K & {f}-Ideal is prime &
not (g in {f}-Ideal) holds {f,g}-Ideal = the carrier of Polynom-Ring K;
theorem :: ALGNUM_1:50
for f,g be Element of Polynom-Ring K holds
f <> 0.Polynom-Ring K & {f}-Ideal is prime &
not g in {f}-Ideal implies {f}-Ideal,{g}-Ideal are_co-prime;
theorem :: ALGNUM_1:51
for x be Element of F_Complex,a be Element of FQ_Ring(x)
ex g be Element of Polynom-Ring F_Rat st a = hom_Ext_eval(x,F_Rat).g;
theorem :: ALGNUM_1:52
for x,a be Element of F_Complex st a <> 0.F_Complex &
a in the carrier of FQ_Ring(x)
ex g be Element of Polynom-Ring F_Rat
st not g in Ann_Poly(x,F_Rat) & a = hom_Ext_eval(x,F_Rat).g;
theorem :: ALGNUM_1:53
for x,a be Element of F_Complex st x is algebraic & a <> 0.F_Complex &
a in the carrier of FQ_Ring(x)
ex f,g be Element of Polynom-Ring F_Rat
st {f}-Ideal = Ann_Poly(x,F_Rat) & not(g in Ann_Poly(x,F_Rat)) &
a = hom_Ext_eval(x,F_Rat).g &
{f}-Ideal,{g}-Ideal are_co-prime;
theorem :: ALGNUM_1:54
for x,a be Element of F_Complex st x is algebraic & a <> 0.F_Complex &
a in the carrier of FQ_Ring(x) holds
ex b be Element of F_Complex st b in the carrier of FQ_Ring(x)
& a*b = 1.F_Complex;
theorem :: ALGNUM_1:55
for x be Element of F_Complex st x is algebraic holds FQ_Ring(x) is Field;